Three Dimensional Geometry — Lesson
1) Hook — Real-Life Story to Grab Attention
Imagine you are an architect designing the iconic Lotus Temple in Delhi. To accurately model its petal-like structures, you need to understand how points, lines, and planes exist in space — not just on a flat surface. This is where Three Dimensional Geometry comes into play, helping you visualize and calculate distances and angles in 3D space, essential for creating safe and stunning architectural marvels.
2) Core Concepts — Understanding 3D Geometry
Three Dimensional Geometry studies points, lines, and planes in space defined by three axes: x, y, and z. Each point is represented as (x, y, z).
- x-axis: Horizontal axis (left-right)
- y-axis: Horizontal axis (front-back)
- z-axis: Vertical axis (up-down)
Example 1: Point in 3D Space
The point P(3, -2, 5) lies 3 units along x-axis, 2 units backward along y-axis, and 5 units upward along z-axis.
Distance Between Two Points
Given points A(x₁, y₁, z₁) and B(x₂, y₂, z₂), the distance between them is:
Direction Ratios and Direction Cosines
For a line with direction ratios (a, b, c), the direction cosines are:
These satisfy: l² + m² + n² = 1
Equation of a Line in 3D
Two common forms:
| Form | Equation | Description |
|---|---|---|
| Symmetric form | (x - x₁)/a = (y - y₁)/b = (z - z₁)/c | Passes through (x₁, y₁, z₁) with direction ratios (a, b, c) |
| Parametric form | x = x₁ + at, y = y₁ + bt, z = z₁ + ct | t is a parameter |
Equation of a Plane
The general form of a plane with normal vector (A, B, C) passing through point (x₀, y₀, z₀) is:
Or in simplified form: Ax + By + Cz + D = 0
Angle Between Two Planes
If planes have normal vectors n₁ = (A₁, B₁, C₁) and n₂ = (A₂, B₂, C₂), then the angle θ between planes is:
Distance from a Point to a Plane
For point P(x₁, y₁, z₁) and plane Ax + By + Cz + D = 0, the perpendicular distance is:
3) Key Formulas / Rules
- Distance Between Two Points:
d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] - Direction Cosines:
l = a / √(a² + b² + c²), m = b / √(a² + b² + c²), n = c / √(a² + b² + c²) - Equation of a Line (Symmetric Form):
(x - x₁)/a = (y - y₁)/b = (z - z₁)/c - Equation of a Plane:
A(x - x₀) + B(y - y₀) + C(z - z₀) = 0 - Angle Between Two Planes:
cos θ = |A₁A₂ + B₁B₂ + C₁C₂| / (√(A₁² + B₁² + C₁²) × √(A₂² + B₂² + C₂²)) - Distance from Point to Plane:
d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)
4) Did You Know?
In 3D geometry, the concept of direction cosines was first used by the great Indian mathematician Bhāskara II in the 12th century, centuries before it became standard in modern mathematics! His work on spherical trigonometry laid foundations that are still relevant in 3D coordinate geometry today.
5) Exam Tips — Common Mistakes & Board Exam Patterns
- Common Mistakes:
- Mixing up direction ratios and direction cosines — remember direction cosines are normalized.
- For distance formulas, forgetting to take the absolute value in numerator.
- Incorrect substitution of point coordinates in plane or line equations.
- Ignoring the sign when calculating angles between planes — use absolute value for cosine.
- Board Exam Patterns:
- Questions on finding distance between points, line-plane intersections, and angle between planes are frequent.
- Derivation or proof-based questions on direction cosines and distance formulas may appear.
- Application-based problems involving real-life 3D objects (e.g., towers, buildings) are common.
- Ensure stepwise solutions with clear substitution and formula application for full marks.
- Tip: Practice previous year questions from CBSE Sample Papers and NCERT Exemplar for 3D geometry to gain confidence.
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